Solutions to problems from section 1.3
4. Let
u
=
bracketleftbigg
3
2
bracketrightbigg
and
v
=
bracketleftbigg
2

1
bracketrightbigg
. Display the following vectors using
arrows on an
xy
graph:
u
,
v
,

v
,

2v
,
u
+
v
,
u

v
,
and
u

2v
.
Solution:
6. Write a linear system which is equivalent to
x
1
bracketleftbigg

2
3
bracketrightbigg
+
x
2
bracketleftbigg
8
5
bracketrightbigg
+
x
3
bracketleftbigg
1

6
bracketrightbigg
=
bracketleftbigg
0
0
bracketrightbigg
Solution:

2
x
1
+ 8
x
2
+
x
3
= 0
,
3
x
1
+ 5
x
2

6
x
3
= 0
.
10. Write a vector equation that is equivalent to the following linear sys
tem:
4
x
1
+
x
2
+ 3
x
3
= 9
,
x
1

7
x
2

2
x
3
= 2
,
8
x
1
+ 6
x
2

5
x
3
= 15
.
Solution:
x
1
4
1
8
+
x
2
1

7
6
+
x
3
3

2

5
=
9
2
15
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12. Is
b
a linear combination of
a
1
,
a
2
, and
a
3
where
a
1
=
1

2
2
,
a
2
=
0
5
5
,
a
3
=
2
0
8
,
b
=

5
11

7
?
Solution:
Answering this question is the same as answering whether
or not the linear system with augmented matrix
1
0
2

5

2
5
0
11
2
5
8

7
is consistent. Let’s row reduce.
1
0
2

5

2
5
0
11
2
5
8

7
NR
2
=
R
2
+2
R
1
→
1
0
2

5
0
5
4
1
2
5
8

7
NR
3
=
R
3

2
R
1
→
1
0
2

5
0
5
4
1
0
5
4
3
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 Fall '08
 AXENOVICH
 Linear Algebra, Vectors, Vector Space, SEPTA Regional Rail, Linear combination

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